Knotty knits are tangles on tori

Abstract

In this paper we outline a topological framework for constructing 2-periodic knitted stitches and an algebra for joining stitches together to form more complicated textiles. Our topological framework can be constructed from certain topological "moves" which correspond to "operations" that knitters make when they create a stitch. In knitting, unlike Jacquard weaves, a set of $n$ loops may be combined in topologically nontrivial ways to create $n$ stitches that are not pairwise associated. We define a \emph{swatch} as a construction that allows for these knitable knots.